DESIGN OF LIQUID RETAINING CONCRETE STRUCTURES
average strain in the concrete and less the tension stiffening provided by the concrete between the cracks. Whereas, in Expression M1, the average strain in the steel is equal to the pure tension strain in the steel, accounting for tension stiffening, less the average strain in the concrete at the surface (all resulting from the imposed deformations that arise from shrinkage and/or change in temperature). The two expressions are similar except for this important subtle difference. Expression M1 is derived in full along with a complete description of the principles of cracking in Chapter 5. The following breakdown of Expression 7.9 is intended to clarify the calculation of the strain in the steel and concrete and the quantification of the effects of tension stiffening. For the FLEXURAL case (Expression 7.9): (sm − cm) = [s–kt ( fct,eff / p,eff)(1 + ep,eff )] / Es
(3.20)
(sm − cm) = s / Es–[kt ( fct,eff / p,eff)(1 + ep,eff )] / Es
(3.21)
Or where s / Es = average maximum strain in the steel due to moment, calculated on the basis of a cracked section. Further expansion of the second term in (3.21) gives: [kt ( fct,eff / p,eff )] / Es + [(kt ( fct,eff / p,eff ))(ep,eff )] / Es
(3.22)
Reducing the second term in (3.22), using e = Es/Ec gives: kt fct,eff / Ec So (3.22) becomes: [kt ( fct,eff / p,eff )] / Es + kt fct,eff / Ec
(3.23)
where the second term of Eq. (3.23) equals the strain in the concrete between the cracks (depending on the duration of the load). By comparing the first term of (3.23) with Equation 13 of BS 8110 Part 2 or the theory presented by Beeby (1979) where tension stiffening correction at the level of the reinforcement is represented as: ∆ = K [( ftfscr ) / (Esft )]
(3.24)
here fscr (steel stress at cracking) = f1 (steel stress under load considered), it is clear that the first term of (3.23) represents tension stiffening. Therefore, for the FLEXURAL case, cm is represented by the second term of Eq. (3.23) and sm is represented by the first term of Eq. (3.21) (average maximum strain in the steel due to applied moment) less the first term of Eq. (3.23) (effect of tension stiffening provided by the concrete between the cracks). So the average strain in the steel is equivalent to the flexural strain in the steel at the crack calculated on the basis of a cracked section less the average strain in the concrete and less the tension stiffening provided by the concrete between the cracks. Whereas, for a pure AXIAL tension case (Expression M1): (sm − cm) = 0.5e kckfct,eff (1+1/e ) / Es
(3.25)
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